The Acoustic Design of Soundproofing Doors and Windows

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1 3 The Open Acoustics Journl, 1, 3, 3-37 The Acoustic Design of Soundproofing Doors nd Windows Open Access Nishimur Yuy,1, Nguyen Huy Qung, Nishimur Sohei 1, Nishimur Tsuyoshi 3 nd Yno Tkshi 1 Kummoto Ntionl College of Technology, 659- Suy Koshi shi Kummoto, Jpn Fculty of Engineering, Kummoto University, Kurokmi Kummoto, Jpn 3 Fculty of Computer Science, Sojo University, 4--1 Iked Kummoto, Jpn Astrct: A model for mnufcturing doors nd ows which re cple of nturl ventilting, reducing trffic noise nd so on for the developing tropicl countries is presented. These ows nd doors comine two sic components which re ventiltion unit nd lighting unit. Due to the fct tht the ventiltion unit must hve lrge volume to ttenute low frequency noise, mny resonnce of higher-order mode wve will e generted inside the unit. In this work, method to predict the insertion-loss of rectngulr ventiltion unit with input nd output openings t vrious positions is proposed y solving the wve eqution, considering the resonnce frequencies of higher-order mode. The results of the nlysis hve een confirmed y experiments. Keywords: Soundproofing, resonnce frequency, csement ows, higher-order modes. 1. INTRODUCTION Seling up types doors nd ows re widely used in the current houses to intercept n inside outside. Needless to sy, some equipments such s ir conditioners, re necessry to keep comfortle indoor temperture while such doors nd ows re closed nd the using durtion of such equipments is often limited ecuse of the power consumption cost nd helth. The uthors hve een presented concept for mnufcturing ows which re cple of ventilting, regulting sunlight nd reducing trffic noise for the developing tropicl countries [1]. These ows comine two sic components which re ventiltion unit nd lighting unit. The former lso serves s n import function in reducing noise. Due to the fct tht the ventiltion unit must hve lrge volume to ttenute low-frequency noise, mny resonnce of higher-order mode wve will e generted inside the unit [- 5]. Consequently, it is necessry to tke into considertion the selection of size nd plcement of input nd output openings in such wy tht would minimize the effects of higher-order mode in order to hve gret soundproofing effect. In the previous nlysis, the ventiltion unit is constructed using rectngulr cvity with input nd output openings t oth ends. Actully, n input nd output cn e locted t vrious positions on the unit ccording to the door or ow s design. Moreover, the optimum loction n insertion loss effect is effectively chieved is still n unknown prolem. In this rticle, method to predict the insertion loss of rectngulr cvity with input nd output openings t vrious positions is proposed. Following the development of the Address correspondence to this uthor t the Kummoto Ntionl College of Technology, 659- Suy Koshi shi Kummoto, Jpn; Tel: ; Fx: ; E-mil: nisimur@cis.sojo-u.c.jp theory in section 3, this is compred to experimentl results in section 4, n excellent greement is otined.. DESIGN OF SOUNDPROOFING WINDOWS/ DOORS AND THE VENTIATION UNIT Csement ows nd doors re consisted of one or severl wooden frmes tht cn e opened nd closed t vrious ngles nd re widely used in countries with tropicl climtes. An exmple of csement door s structure is shown in Fig. 1). The ows nd doors re opened during dy for nturlly ventilting the room nd closed t night, even when closed room ventiltion is still chieved ecuse the ows nd doors re constructed with lots of ventilting slits. However, n nnul increse in trffic noise level these countries hve rendered these ows nd doors to e useless ecuse these ventilting slits serve s direct pthwy for trffic noise to enter the home. hinges ventiltion slits hinges outside inside Fig. 1). The csement doors of current houses in tropicl climte countries /1 1 Benthm Open

2 The Acoustic Design of Soundproofing Doors nd Windows The Open Acoustics Journl, 1, Volume 3 31 Bsed on the previous reports, our proposed csement s door is shown in Fig. ). It comines two sic components: ventiltion nd lighting. The lighting unit cn e constructed using one or two glss lyers which re mounted etween two ventiltion components with input nd output openings s shows in Fig. 3). The ventiltion unit my consist of squre, rectngulr or more complicted shpes depending on decortive considertions. Needless to sy, the unit requires simple internl structure nd lrge input nd output to mximize ventiltion, s well s preventing outside noise from entering the home. Actully, n input nd output cn e locted t vrious positions on the cvity ccording to the door or ow design. Therefore, these comintions re considered s follows: exchnging coordintes. We nme c) nd d) s Model-. The sound propgtion in ) is the sme s c) nd tht in ) is the sme s d). ) ) hinges c) d) lighting unit Fig. 4). Comintions of input nd output positions on rectngulr cvity. ventiltion unit 3. METHOD OF ANAYSIS 3.1. Insertion oss Acoustic chrcteristics of n coustic element cn e descried y the four-pole prmeters A, B, C nd D s [6] Fig. ). The proposed csement doors. hinges P 1 U 1 A C = B D P U = cos kl j 1 sin kl Z j Z sin kl cos kl P U 1) P 1 nd U 1 re the sound pressure nd velocity t the input, P nd U re t the output, Z is the chrcteristic impednce, k is the wve numer nd l is the length. When this element is connected to the source, their performnce cn e expressed through the use of insertion loss I defined y [6] ) ) c) d) Fig. 3). Design of the proposed csement ows. Comintions of input nd output positions re shown in Fig. 4) in which ) nd ) re the cses they re in opposite fces while, c) nd d) re in fces which crossed right ngles. Difference etween ) nd ) is whether they re locted in ig or smll cvity re, however it is quite similr in the theoreticl nlysis. Herefter, we nme ) nd ) s Model-1. Similrly, difference etween c) nd d) is whether they re locted in ig or smll side re which crossed right ngles. In this nlysis, once we find the coustic chrcteristic when the input nd output re locted in ritrry fce, it is esily computle for other fces y I = 1 log W r W = 1 log U 1 U Here, W r nd W re the rdited power t one point in spce with or without the coustic element inserted etween tht point nd the source. The rtio of U 1 U is equl to the D prmeter of Eq. 1), s fr s constnt velocity source is concerned. When the coustic element is connected in series s shown in Fig. 5), the sectionl re of element 1 nd 3 ecomes significntly smll s compred to the sectionl re of element nd the D prmeter of whole system cn e descried y the following pproximte eqution D = cos kl 1 ) C w ) jz 3 sin kl 3 ) 3) )

3 3 The Open Acoustics Journl, 1, Volume 3 Yuy et l. C w denoted the C prmeter of element. As shown in Eq. ) nd 3), in order to otin relile I effect, D prmeter must e high enough. In other words, the design of element- to hve high enough prmeter C w is demnded. l l1 l 3 [3] t y =, V y = 7) [4] t y =, V y = 8) [5] t z =, V z = V i F i x, y) 9) [6] t z =, V z = 1) V i is the driving velocity t the input, F i x, y) is the function which hs vlue of 1 t the input section re nd t the other re, nmely x i i x x i i, y i i y y i i ) Fig. 5). Bsic structure of ventiltion unit. x At first, Eq. 5) gives C sin x D cos x ) And from Eq. 6) x= = C = 11) 3.. Computtion of C w First of ll, we find the sound pressure on the input nd output of the Model-1 s shown in Fig. 6). The input nd output hve sectionl re of S i = i i nd = its center is locted t point I x i, y i, ) nd O x, y, ), respectively. y z Fig. 6). Model of computtion. The complete wve-eqution in terms of the velocity potentil when expressed in rectngulr coordintes is given y [1] α i I ) α β i O β x C sin x D cos x = Ae µz Be µz E sin s y F cos s y ) 4) A, B, C, D, E nd F re ritrry constnts determinle from the oundry conditions, α, s nd µ re the constnts. et V x = / x, V y = / y nd V z = / z e the velocity components in the x, y nd z directions, respectively. Assuming the wlls of the cvity to e perfectly rigid nd the loss t the wll cn e neglected, the oundry conditions re [1] t x =, V x = 5) [] t x =, V x = 6) y z α α i I ) O β i β x x D cos x = = m m =, 1,,.) 1) x= Sustituting Eqs. 11) nd 1) into Eq. 4) gives = Ae µz Be µz cos m ) m= E sin s y F cos s y ) 13) from Eq. 7) E sin s y F cos s y ) y nd from Eq. 8) F cos s y y y= = s = n y= = E = 14) n =, 1,,.) 15) Sustituting Eqs. 14) nd 15) into Eq. 13), so ecomes = Ae z Be z cos m x ) cos n y ) 16) = m / ) n / ) k 17) z Next, from Eq. 1 ) Ae z Be z z= Sustituting B into Eq. 16) yields = B = Ae 18) = e cosh z ) ) C m,n cos m x ) cos n y ) 19) from Eq. 9)

4 The Acoustic Design of Soundproofing Doors nd Windows The Open Acoustics Journl, 1, Volume 3 33 z cos n y ) C m,n cos m x e cosh z ) z= = V i F i x, y) ) ) By multiplying oth sides of Eq. ) y cosm x / )cosn y / ) nd integrting with respect to x from to nd with respect to y from to, C m,n cn e determined. Sustitution of C m,n to 19) yields cos m x cosh z ) = sinh ) cos n y ) 1) 16 = cos m x i m n S i cos n y i sin = V i S i is the volume velocity t input. m i sin n) i ) Therefore, sound pressure corresponding to ecomes Px, y, z) = jkc cosh z ) S w cos m x sinh ) cos n y ) 3) Z w = c / S w represents the chrcteristic coustic impednce of the cvity. The sound pressure t the input side z = ecomes P i = Px, y, ) = cosh j kz w S w cos m x sinh ) cos n y ) 4) Therefore, the verge sound pressure on the input is given y x i i y i i P i = 1 P x, y, ) S i dx dy x i i y i i E m,n cosh 5) sinh the level E m,n corresponding to m, n) modes is given s E m,n = S x w cos m x S i cos n y i i y i ) i dx dy 6) x i i y i ) i Similrly, y setting z= in Eq. 3), the sound pressure on the output ecomes P = Px, y, ) = 1 j kz w S w cos m x sinh ) cos n y ) 7) nd the verge sound pressure on the output ecomes P = 1 x x y y Px, y, )dx dy Q = jkz w U mn sinh ) S x y w m x n y m, n m, n cos cos S x y Q = D dx dy ) ) 8) 9) Moreover, expnding Eq. 8) corresponding to the vrition of m nd n, we hve 1 P = j Z w sin k Q m,n sinh ) Q m, = S w Q,n = S w µ m, sinh µ m, Q m, m=1 n=1 Q,n µ,n sinh µ,n 3) 4 m D cos m x m, sin m) 31) 4 n D cos n y,n sin n) 3) 8 D m, = i m S i cos m x i sin m) i 33) D,n = 8 i n S i cos n y i sin n) i 34) the symol mens without the cses of m nd n in 1st 3rd term of Eq. 3). The C w cn e otin y C w = / P under the condition of the output velocity of the volume, U = V z is equl to zero. Note tht, this condition hs lredy een considered in Eq. 1). In the cse of Model- s shown in Fig. 6), the input nd output hve sectionl re of S i = i i nd = its center is locted t point I x i, y i, ) nd O x,, z ), respectively. After the similr computtion procedure is performed from Eq. 4) to Eq. 3), the sound pressure on the side the output locted cn e derived s cosh z ) P = Px,, z) S w sinh cos m x ) cos n 35)

5 34 The Open Acoustics Journl, 1, Volume 3 Yuy et l. Therefore, the verge output sound pressure ecomes x z P = 1 P x,, z) dz dy x z S U w cos n i sinh x x z ) z ) cosh z ) )cos m x dz dx Rm,n = jkz w 36) sinh Rm,n = S w cos n x z ) cosh µ z ) m,n x z ) cos m x 37) dz dx Moreover, expnding the ove eqution corresponding to vrition of m nd n, we hve R, P = j Z w k sin k R,n µ,n sinh µ,n n=1 R m, m=1 µ m, sinh µ m, R m,n sinh ) R, = 1 x z cos kz ) )dz dx = 1 k x z 38) { sink z ) sink z ) } 39) Finlly, C w cn e derive s C w = P 4) 4. RESUT AND DISCUSSION C w is defined y Eq. 4) including the verge output sound pressure P t its denomintor, P is given in Eq. 3) nd Eq. 38) for Model-1 nd Model-, respectively. Note tht oth models hve the sme verge input sound pressure P i defined y Eq. 5). In order to otin n I effectively C w must e t gret, in other words, low level of P is preferle. Referring to Eq. 3), P ecomes gret when its denomintor sink) nd sinh resonnce frequencies of re zero, nmely, t the following sink) = f = c sinh ) = f m,n = c =, 1,, 3... ) = 1,, 3... ) 41) m n ) 4) The f represents the resonnce frequencies of the plne wve nd f m,n represents those of trverse wve. The first resonnce frequencies of f m,n when = occur sequentilly, re shown in Tle 1, when the dimensions of the cvity re =.48m, =.75m nd =.9m, respectively. Generlly, t the frequency rnge the trverse wves re generted, the sound pressure P will increse nd when the resonnce frequencies of other modes co-occur, C w will e smll nd the I cn not e expected to e s gret. Genertion mechnism of these frequencies cn e understood ccording to the clcultion exmple s shown in Fig. 7) with,) nd 4,) modes. Dimensions of the cvity used in clcultion re the sme s in Tle 1. Tle 1. The First Resonnce Frequency of Some m,n) Mode Occur in the Cvity when =.48m, =.75m nd =.9m Mode 1st Resonnce m n Frequency Hz) Sound pressure level of those wve components in db re shown in Fig. 7). They lso hve mny resonnce frequencies which occur corresponding to the incresing of in Eq. 4). Therefore, it is cler tht when we eliminte n ritrry higher-order wve mode y ny method, we will not only void mny resonnces generted y this mode ut lso otin the low level of the entire output sound pressure.

6 The Acoustic Design of Soundproofing Doors nd Windows The Open Acoustics Journl, 1, Volume 3 35 Fig. 7). Physicl mening of Eq. 3). ) Clculted exmple of Eq. 3) with,) mode nd 4,) mode. Resonnces will occur when these denomintors ecome zero. Dimensions of cvity re =.48m, =.75m nd =.9m. ) Spectrum of,) mode, 4,) mode. Next, in order to exmine the ccurcy of our predicted result, the mesurement in some positions on the cvity ws crried out y using the previous mesurement method see Appendix). Mesurement result of C w of Model-1 when the input nd output re locted in the center position /, /) is shown in Fig. 8), while the denomintors of Eq. 3) re shown in Fig. 8). Agreement oserved etween the mesurement nd our predicted resonnce frequencies is cceptle. Note tht, y locting the input or output t x=/ nd y=/, cos m x / ecome cos m / cos n y / ) in Eq. 7) will cos n / ) nd s result, the m, n) mode sound pressure does not pper when m=1, 3, 5 for ll vlues of n nd when n=1, 3, 5. for ll vlues of m. Therefore, s shown in Fig. 8), the m, n) mode sound pressures ppers in cvity in the order of, ), 4, ) nd so on. Fig. 9) shows the mesured nd theoreticl results of Model- t point A /,, /) locted on the top of the cvity. By locting t z=/, Eq. 35) ecomes = P /,, / ) = cosh / j kz w S w cos m sinh ) cos n S w 1 cos m ) cos n 1 sinh / ) 43) Fig. 8). Resonnce frequencies pper in the cvity when the output is locted t position E. ) computed result of the denomintor of Eq. 3). ) mesured result of C w. Moreover, expnding the ove eqution with mode,) we hve S w 1 ) 1 k sink / ) 1 sinh / cos m cos n ), -. / 44) As result, the denomintors of ecome sink / ) nd sinh /, therefore, the resonnce frequencies will differ from those of Fig. 8). Clcultion exmples of mode,) nd mode4,) re shown in Fig. 9). The frequencies t which sink / ) nd ecome zero correspond well to those of the sinh / mesured results shown in Fig. 9). In order to show the difference mong the resonnce frequencies, the mesured result t position E, which hd een descried ove, ws dded to Fig. 9). Fig. 1) shows the mesured result t point C, /, /) tht loctes in the right side of the cvity. Becuse the resonnce frequencies re the sme s point A s descried ove, the effectiveness of our clcultion method hs een proven.

7 36 The Open Acoustics Journl, 1, Volume 3 Yuy et l. locted on the ig cvity re, resonnce frequencies of higher-order mode generlly pper in low frequency rnge. Fig. 9). Resonnce frequencies pper in the cvity when the output is locted t position A nd E. ) computed result of the denomintor of Eq. 44) ) mesured result of C w. Fig. 11). Resonnce frequencies pper in the cvity when the output is locted t position A with chnging input loction ) computed result of the denomintor of Eq. 3). ) mesured result of C w. Tle. The First Resonnce Frequency of Some m,n) Mode Occur in the Cvity when =.48m, =.9m nd =.75m Mode 1 st Resonnce m n Frequency Hz) Fig. 1). Mesured results t position A nd C. Fig. 11) shows the theoreticl nd mesured results of Model-1 t point A /,, /) which loctes on the top of the cvity when the input locted on the opposite fce. In this cse, the first resonnce frequencies of f m,n tht occur sequentilly re shown in Tle. Becuse the input ws

8 The Acoustic Design of Soundproofing Doors nd Windows The Open Acoustics Journl, 1, Volume CONCUSIONS The chrcteristic of sound propgtion in the rectngulr ventiltion hving n input nd output locted t vrious positions, hs een presented y solving the wve eqution, considering the higher-order mode effects. Bsed on the otined results, the cuse nd mechnism of resonnce frequencies of prmeter C re discussed in detil. To prove the theory, experiments with vrious positions were crried out nd excellent greement is otined. The formuls derived from the present study enle the ccount of the insertion loss of the ventiltion in prcticl pplictions. Also, it will e suitle for further study on the determintion of optimized positions of input nd output. This technology will e presented in n upcoming report. APPENDIX A lock digrm of the whole experimentl pprtus to mesure the prmeter C w is shown in Fig. 1) [7]. Two microphones were locted t oth sides of the ventiltion unit to mesure the sound pressures nd P B. Reltionship etween nd P B is given y U A cos kl j Z sin kl = j 1 sin kl cos kl Z A w C w B w D w P B U B A-1) the first term nd the second term represent the fourpole prmeters of the input pipe which hs length l nd the ventiltion unit, respectively. Symols U A, U B nd Z represent velocity of the volume nd coustic impednce of the input pipe. By instlling the microphone on the output piston, U B will ecome zero, therefore Eq. A-1) cn e written s P B A-) = A w cos kl C w Z sin kl Furthermore, since the first term cn e disregrded s compred to the second term, Eq. A-) cn e written in terms of db s follows og 1 C w = og 1 / P B og 1 Z sin kl A-3) This mens C w cn e otined y sutrcting the coustic chrcteristic of the input pipe Z sin kl from the mesured sound pressures nd P B. Fig. 1). Block digrm of the experimentl pprtus. REFERENCES [1] Nishimur Y, Nishimur S, Nishimur T, Yno T. Sound propgtion in soundproofing csement ows. J Appl Acoust 9; 79): [] Ih J. G. The rective ttenution of rectngulr plenum chmers. J Sound Vi 199; 1571): [3] Eriksson J. Higher order mode effects in circulr ducts nd expnsion chmers. J Acoust Soc Am 198; 68): [4] Hnsen Z. Active control of higher order coustic modes in ducts. J Acoust Soc Am 199; 91): [5] Yin Y, Horoshenkov. The ttenution of the higher-order crosssection modes in duct with thin porous lyer. J Acoust Soc Am 5; 117): [6] Munjl M. Acoustics of Ducts nd Mufflers. New York: Willey [7] Nishimur S, Nishimur T, Yno T. Acoustic nlysis of ellipticl muffler chmer hving perforted pipe. J Sound Vi 6; 97: Received: Decemer 15, 9 Revised: Mrch 9, 1 Accepted: June 3, 1 Yuy et l.; icensee Benthm Open. This is n open ccess rticle licensed under the terms of the Cretive Commons Attriution Non-Commercil icense which permits unrestricted, non-commercil use, distriution nd reproduction in ny medium, provided the work is properly cited.